singular value decompositions \(\mathbf{A} = \mathbf{U} \Sigma \mathbf{V}^T\)
iterative methods for numerical linear algebra
Except for the iterative methods, most of these numerical linear algebra tasks are implemented in the BLAS and LAPACK libraries. They form the building blocks of most statistical computing tasks (optimization, MCMC).
Our major goal (or learning objectives) is to
know the complexity (flop count) of each task
be familiar with the BLAS and LAPACK functions (what they do)
do not re-invent wheels by implementing these dense linear algebra subroutines by yourself
understand the need for iterative methods
apply appropriate numerical algebra tools to various statistical problems
All high-level languages (Julia, Matlab, Python, R) call BLAS and LAPACK for numerical linear algebra.
Julia offers more flexibility by exposing interfaces to many BLAS/LAPACK subroutines directly. See documentation.
2 BLAS
BLAS stands for basic linear algebra subprograms.
See netlib for a complete list of standardized BLAS functions.
Matlab uses Intel’s MKL (mathematical kernel libaries). MKL implementation is the gold standard on market. It is not open source but the compiled library is free for Linux and MacOS. However, not surprisingly, it only works on Intel CPUs.
Julia uses OpenBLAS. OpenBLAS is the best cross-platform, open source implementation. With the MKL.jl package, it’s also very easy to use MKL in Julia.
Typical BLAS functions support single precision (S), double precision (D), complex (C), and double complex (Z).
3 Examples
The form of a mathematical expression and the way the expression should be evaluated in actual practice may be quite different.
Some operations appear as level-3 but indeed are level-2.
Example 1. A common operation in statistics is column scaling or row scaling \[
\begin{eqnarray*}
\mathbf{A} &=& \mathbf{A} \mathbf{D} \quad \text{(column scaling)} \\
\mathbf{A} &=& \mathbf{D} \mathbf{A} \quad \text{(row scaling)},
\end{eqnarray*}
\] where \(\mathbf{D}\) is diagonal. For example, in generalized linear models (GLMs), the Fisher information matrix takes the form \[
\mathbf{X}^T \mathbf{W} \mathbf{X},
\] where \(\mathbf{W}\) is a diagonal matrix with observation weights on diagonal.
Column and row scalings are essentially level-2 operations!
usingBenchmarkTools, LinearAlgebra, RandomRandom.seed!(257) # seedn =2000A =rand(n, n) # n-by-n matrixd =rand(n) # n vectorD =Diagonal(d) # diagonal matrix with d as diagonal
# This works only when Matlab is installedusingMATLABmat"""d = rand(5, 1)diag(d)"""
ArgumentError: ArgumentError: Package MATLAB not found in current path.
- Run `import Pkg; Pkg.add("MATLAB")` to install the MATLAB package.
Example 2. Innter product between two matrices \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{m \times n}\) is often written as \[
\text{trace}(\mathbf{A}^T \mathbf{B}), \text{trace}(\mathbf{B} \mathbf{A}^T), \text{trace}(\mathbf{A} \mathbf{B}^T), \text{ or } \text{trace}(\mathbf{B}^T \mathbf{A}).
\] They appear as level-3 operation (matrix multiplication with \(O(m^2n)\) or \(O(mn^2)\) flops).
Random.seed!(123)n =2000A, B =randn(n, n), randn(n, n)# slow way to evaluate tr(A'B): 2mn^2 flops@benchmarktr(transpose($A) *$B)
BenchmarkTools.Trial: 95 samples with 1 evaluation.
Range (min … max): 50.924 ms … 59.158 ms┊ GC (min … max): 0.00% … 3.94%
Time (median): 52.136 ms ┊ GC (median): 0.00%
Time (mean ± σ): 52.486 ms ± 1.530 ms┊ GC (mean ± σ): 0.86% ± 1.46%
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50.9 ms Histogram: frequency by time 58.3 ms <
Memory estimate: 30.52 MiB, allocs estimate: 2.
But \(\text{trace}(\mathbf{A}^T \mathbf{B}) = <\text{vec}(\mathbf{A}), \text{vec}(\mathbf{B})>\). The latter is level-1 BLAS operation with \(O(mn)\) flops.
# smarter way to evaluate tr(A'B): 2mn flops@benchmarkdot($A, $B)
BenchmarkTools.Trial: 2732 samples with 1 evaluation.
Range (min … max): 1.649 ms … 1.984 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 1.823 ms ┊ GC (median): 0.00%
Time (mean ± σ): 1.827 ms ± 31.177 μs┊ GC (mean ± σ): 0.00% ± 0.00%
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1.65 ms Histogram: frequency by time 1.93 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
Example 3. Similarly \(\text{diag}(\mathbf{A}^T \mathbf{B})\) can be calculated in \(O(mn)\) flops.
# slow way to evaluate diag(A'B): O(n^3)@benchmarkdiag(transpose($A) *$B)
BenchmarkTools.Trial: 93 samples with 1 evaluation.
Range (min … max): 50.993 ms … 100.033 ms┊ GC (min … max): 0.00% … 1.47%
Time (median): 52.402 ms ┊ GC (median): 0.00%
Time (mean ± σ): 53.965 ms ± 6.802 ms┊ GC (mean ± σ): 0.90% ± 1.25%
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51 msHistogram: log(frequency) by time 88.9 ms <
Memory estimate: 30.53 MiB, allocs estimate: 3.
# smarter way to evaluate diag(A'B): O(n^2)@benchmarkDiagonal(vec(sum($A .*$B, dims =1)))
BenchmarkTools.Trial: 1294 samples with 1 evaluation.
Range (min … max): 3.305 ms … 5.589 ms┊ GC (min … max): 0.00% … 23.28%
Time (median): 3.625 ms ┊ GC (median): 0.00%
Time (mean ± σ): 3.860 ms ± 513.706 μs┊ GC (mean ± σ): 8.86% ± 10.65%
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3.3 ms Histogram: frequency by time 4.88 ms <
Memory estimate: 30.53 MiB, allocs estimate: 5.
To get rid of allocation of intermediate arrays at all, we can just write a double loop or use dot function.
functiondiag_matmul!(d, A, B) m, n =size(A)@assertsize(B) == (m, n) "A and B should have same size"fill!(d, 0)for j in1:n, i in1:m d[j] += A[i, j] * B[i, j]endDiagonal(d)endd =zeros(eltype(A), size(A, 2))@benchmarkdiag_matmul!($d, $A, $B)
BenchmarkTools.Trial: 1479 samples with 1 evaluation.
Range (min … max): 3.117 ms … 3.586 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 3.372 ms ┊ GC (median): 0.00%
Time (mean ± σ): 3.379 ms ± 41.213 μs┊ GC (mean ± σ): 0.00% ± 0.00%
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3.12 ms Histogram: frequency by time 3.5 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
Exercise: Try @turbo (SIMD) and @tturbo (multi-threaded SIMD) from LoopVectorization.jl package.
4 Memory hierarchy and level-3 fraction
Key to high performance is effective use of memory hierarchy. True on all architectures.
Flop count is not the sole determinant of algorithm efficiency. Another important factor is data movement through the memory hierarchy.
In Julia, we can query the CPU topology by the Hwloc.jl package. For example, this laptop runs an Apple M2 Max chip with 4 efficiency cores and 8 performance cores.
For example, Xeon X5650 CPU has a theoretical throughput of 128 DP GFLOPS but a max memory bandwidth of 32GB/s.
Can we keep CPU cores busy with enough deliveries of matrix data and ship the results to memory fast enough to avoid backlog?
Answer: use high-level BLAS as much as possible.
A distinction between LAPACK and LINPACK (older version of R uses LINPACK) is that LAPACK makes use of higher level BLAS as much as possible (usually by smart partitioning) to increase the so-called level-3 fraction.
To appreciate the efforts in an optimized BLAS implementation such as OpenBLAS (evolved from GotoBLAS), see the Quora question, especially the video. Bottomline is
Get familiar with (good implementations of) BLAS/LAPACK and use them as much as possible.
5 Effect of data layout
Data layout in memory affects algorithmic efficiency too. It is much faster to move chunks of data in memory than retrieving/writing scattered data.
Storage mode: column-major (Fortran, Matlab, R, Julia) vs row-major (C/C++).
Cache line is the minimum amount of cache which can be loaded and stored to memory.
x86 CPUs: 64 bytes
ARM CPUs: 32 bytes
In Julia, we can query the cache line size by Hwloc.jl.
# Apple Silicon (M1/M2 chips) don't have L3 cacheHwloc.cachelinesize()
ErrorException: Your system doesn't seem to have an L3 cache.
Accessing column-major stored matrix by rows (\(ij\) looping) causes lots of cache misses.
Take matrix multiplication as an example \[
\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}, \quad \mathbf{A} \in \mathbb{R}^{m \times p}, \mathbf{B} \in \mathbb{R}^{p \times n}, \mathbf{C} \in \mathbb{R}^{m \times n}.
\] Assume the storage is column-major, such as in Julia. There are 6 variants of the algorithms according to the order in the triple loops.
jki or kji looping:
# inner most loopfor i in1:m C[i, j] = C[i, j] + A[i, k] * B[k, j]end
- `ikj` or `kij` looping:
# inner most loop for j in1:n C[i, j] = C[i, j] + A[i, k] * B[k, j]end
ijk or jik looping:
# inner most loop for k in1:p C[i, j] = C[i, j] + A[i, k] * B[k, j]end
We pay attention to the innermost loop, where the vector calculation occurs. The associated stride when accessing the three matrices in memory (assuming column-major storage) is
Variant
A Stride
B Stride
C Stride
\(jki\) or \(kji\)
Unit
0
Unit
\(ikj\) or \(kij\)
0
Non-Unit
Non-Unit
\(ijk\) or \(jik\)
Non-Unit
Unit
0
Apparently the variants \(jki\) or \(kji\) are preferred.
""" matmul_by_loop!(A, B, C, order)Overwrite `C` by `A * B`. `order` indicates the looping order for triple loop."""functionmatmul_by_loop!(A::Matrix, B::Matrix, C::Matrix, order::String) m =size(A, 1) p =size(A, 2) n =size(B, 2)fill!(C, 0)if order =="jki"@inboundsfor j =1:n, k =1:p, i =1:m C[i, j] += A[i, k] * B[k, j]endendif order =="kji"@inboundsfor k =1:p, j =1:n, i =1:m C[i, j] += A[i, k] * B[k, j]endendif order =="ikj"@inboundsfor i =1:m, k =1:p, j =1:n C[i, j] += A[i, k] * B[k, j]endendif order =="kij"@inboundsfor k =1:p, i =1:m, j =1:n C[i, j] += A[i, k] * B[k, j]endendif order =="ijk"@inboundsfor i =1:m, j =1:n, k =1:p C[i, j] += A[i, k] * B[k, j]endendif order =="jik"@inboundsfor j =1:n, i =1:m, k =1:p C[i, j] += A[i, k] * B[k, j]endendendusingRandomRandom.seed!(123)m, p, n =2000, 100, 2000A =rand(m, p)B =rand(p, n)C =zeros(m, n);
BenchmarkTools.Trial: 86 samples with 1 evaluation.
Range (min … max): 58.056 ms … 59.729 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 58.433 ms ┊ GC (median): 0.00%
Time (mean ± σ): 58.453 ms ± 234.579 μs┊ GC (mean ± σ): 0.00% ± 0.00%
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58.1 ms Histogram: frequency by time 59.6 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
@benchmarkmatmul_by_loop!($A, $B, $C, "kji")
BenchmarkTools.Trial: 27 samples with 1 evaluation.
Range (min … max): 185.771 ms … 190.199 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 186.726 ms ┊ GC (median): 0.00%
Time (mean ± σ): 187.133 ms ± 1.237 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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186 ms Histogram: frequency by time 190 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
\(ikj\) and \(kij\) looping:
@benchmarkmatmul_by_loop!($A, $B, $C, "ikj")
BenchmarkTools.Trial: 10 samples with 1 evaluation.
Range (min … max): 512.687 ms … 522.756 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 515.130 ms ┊ GC (median): 0.00%
Time (mean ± σ): 515.765 ms ± 2.925 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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513 ms Histogram: frequency by time 523 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
@benchmarkmatmul_by_loop!($A, $B, $C, "kij")
BenchmarkTools.Trial: 10 samples with 1 evaluation.
Range (min … max): 508.427 ms … 513.273 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 509.480 ms ┊ GC (median): 0.00%
Time (mean ± σ): 510.326 ms ± 2.027 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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508 ms Histogram: frequency by time 513 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
\(ijk\) and \(jik\) looping:
@benchmarkmatmul_by_loop!($A, $B, $C, "ijk")
BenchmarkTools.Trial: 20 samples with 1 evaluation.
Range (min … max): 247.799 ms … 257.115 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 249.820 ms ┊ GC (median): 0.00%
Time (mean ± σ): 250.799 ms ± 2.507 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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248 ms Histogram: frequency by time 257 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
@benchmarkmatmul_by_loop!($A, $B, $C, "ijk")
BenchmarkTools.Trial: 21 samples with 1 evaluation.
Range (min … max): 246.205 ms … 256.408 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 249.057 ms ┊ GC (median): 0.00%
Time (mean ± σ): 249.382 ms ± 2.296 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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246 ms Histogram: frequency by time 256 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
Question: Can our loop beat BLAS? Julia wraps BLAS library for matrix multiplication. We see BLAS library wins hands down (multi-threading, Strassen algorithm, higher level-3 fraction by block outer product).
@benchmarkmul!($C, $A, $B)
BenchmarkTools.Trial: 1882 samples with 1 evaluation.
Range (min … max): 2.553 ms … 9.490 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 2.619 ms ┊ GC (median): 0.00%
Time (mean ± σ): 2.655 ms ± 259.992 μs┊ GC (mean ± σ): 0.00% ± 0.00%
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2.55 ms Histogram: frequency by time 2.8 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
# direct call of BLAS wrapper function@benchmarkLinearAlgebra.BLAS.gemm!('N', 'N', 1.0, $A, $B, 0.0, $C)
BenchmarkTools.Trial: 1903 samples with 1 evaluation.
Range (min … max): 2.539 ms … 7.890 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 2.565 ms ┊ GC (median): 0.00%
Time (mean ± σ): 2.626 ms ± 245.462 μs┊ GC (mean ± σ): 0.00% ± 0.00%
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2.54 ms Histogram: frequency by time 2.84 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
Question (again): Can our loop beat BLAS?
Exercise: Annotate the loop in matmul_by_loop! by @turbo and @tturbo (multi-threading) and benchmark again.
6 BLAS in R
Tip for R users. Standard R distribution from CRAN uses a very out-dated BLAS/LAPACK library.
usingRCallR"""sessionInfo()"""
RObject{VecSxp}
R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.4.1
Matrix products: default
BLAS: /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] C
time zone: America/Los_Angeles
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] compiler_4.3.2
┌ Warning: RCall.jl:
│ Attaching package: 'dplyr'
│
│ The following objects are masked from 'package:stats':
│
│ filter, lag
│
│ The following objects are masked from 'package:base':
│
│ intersect, setdiff, setequal, union
│
└ @ RCall /Users/huazhou/.julia/packages/RCall/dDAVd/src/io.jl:172
┌ Warning: RCall.jl: Warning: Some expressions had a GC in every iteration; so filtering is disabled.
└ @ RCall /Users/huazhou/.julia/packages/RCall/dDAVd/src/io.jl:172
# A tibble: 1 x 13
expression min median `itr/sec` mem_alloc `gc/sec` n_itr n_gc
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl> <int> <dbl>
1 A %*% B 125ms 126ms 7.80 30.5MB 7.80 4 4
total_time result memory time
<bch:tm> <list> <list> <list>
1 513ms <dbl [2,000 x 2,000]> <Rprofmem [1 x 3]> <bench_tm [4]>
gc
<list>
1 <tibble [4 x 3]>
Re-build R from source using OpenBLAS or MKL will immediately boost linear algebra performance in R. Google build R using MKL to get started. Similarly we can build Julia using MKL.
Matlab uses MKL. Usually it’s very hard to beat Matlab in terms of linear algebra.
usingMATLABmat"""f = @() $A * $B;timeit(f)"""
ArgumentError: ArgumentError: Package MATLAB not found in current path.
- Run `import Pkg; Pkg.add("MATLAB")` to install the MATLAB package.
7 Avoid memory allocation: some examples
7.1 Transposing matrix is an expensive memory operation
In R, the command
t(A) %*% x
will first transpose A then perform matrix multiplication, causing unnecessary memory allocation